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Properties

Label 2-2061-229.114-c0-0-0
Degree $2$
Conductor $2061$
Sign $0.404 - 0.914i$
Analytic cond. $1.02857$
Root an. cond. $1.01418$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.915 + 0.401i)4-s + (−0.108 + 0.222i)7-s + (−1.29 + 1.52i)13-s + (0.677 + 0.735i)16-s + (−0.156 + 1.88i)19-s + (−0.986 + 0.164i)25-s + (−0.188 + 0.159i)28-s + (0.879 − 0.524i)31-s + (1.19 − 1.29i)37-s + (1.13 − 1.23i)43-s + (0.576 + 0.740i)49-s + (−1.79 + 0.877i)52-s + (0.671 − 1.02i)61-s + (0.324 + 0.945i)64-s + (−0.313 − 1.49i)67-s + ⋯
L(s)  = 1  + (0.915 + 0.401i)4-s + (−0.108 + 0.222i)7-s + (−1.29 + 1.52i)13-s + (0.677 + 0.735i)16-s + (−0.156 + 1.88i)19-s + (−0.986 + 0.164i)25-s + (−0.188 + 0.159i)28-s + (0.879 − 0.524i)31-s + (1.19 − 1.29i)37-s + (1.13 − 1.23i)43-s + (0.576 + 0.740i)49-s + (−1.79 + 0.877i)52-s + (0.671 − 1.02i)61-s + (0.324 + 0.945i)64-s + (−0.313 − 1.49i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312436168\)
\(L(\frac12)\) \(\approx\) \(1.312436168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
229 \( 1 + (-0.614 - 0.789i)T \)
good2 \( 1 + (-0.915 - 0.401i)T^{2} \)
5 \( 1 + (0.986 - 0.164i)T^{2} \)
7 \( 1 + (0.108 - 0.222i)T + (-0.614 - 0.789i)T^{2} \)
11 \( 1 + (0.879 + 0.475i)T^{2} \)
13 \( 1 + (1.29 - 1.52i)T + (-0.164 - 0.986i)T^{2} \)
17 \( 1 + (-0.986 + 0.164i)T^{2} \)
19 \( 1 + (0.156 - 1.88i)T + (-0.986 - 0.164i)T^{2} \)
23 \( 1 + (-0.969 + 0.245i)T^{2} \)
29 \( 1 + (0.614 + 0.789i)T^{2} \)
31 \( 1 + (-0.879 + 0.524i)T + (0.475 - 0.879i)T^{2} \)
37 \( 1 + (-1.19 + 1.29i)T + (-0.0825 - 0.996i)T^{2} \)
41 \( 1 + (0.915 + 0.401i)T^{2} \)
43 \( 1 + (-1.13 + 1.23i)T + (-0.0825 - 0.996i)T^{2} \)
47 \( 1 + (-0.915 + 0.401i)T^{2} \)
53 \( 1 + (0.546 - 0.837i)T^{2} \)
59 \( 1 + (-0.996 - 0.0825i)T^{2} \)
61 \( 1 + (-0.671 + 1.02i)T + (-0.401 - 0.915i)T^{2} \)
67 \( 1 + (0.313 + 1.49i)T + (-0.915 + 0.401i)T^{2} \)
71 \( 1 + (0.879 - 0.475i)T^{2} \)
73 \( 1 + (0.464 + 0.331i)T + (0.324 + 0.945i)T^{2} \)
79 \( 1 + (1.46 - 0.714i)T + (0.614 - 0.789i)T^{2} \)
83 \( 1 + (-0.0825 + 0.996i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.242 + 1.45i)T + (-0.945 - 0.324i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.575139855781675097346777097782, −8.625574474782566288109565376364, −7.64784993230976451724084228594, −7.31695055232488065762969716187, −6.25406315577555213410623749879, −5.76593540702233076946391357994, −4.41641097823377743826437572869, −3.71976873501777117040539536595, −2.44876262748259463338908791063, −1.84999094658854909405985540222, 0.915029607012628005143952406736, 2.55838328377399325428625127968, 2.88400766683527266889385768823, 4.42248084761966292366692997783, 5.24131663263698160179440157243, 6.03716334995444883830397853307, 6.90584233238602558623085380130, 7.50655508978533944521562440146, 8.215045048606452638941970166716, 9.343631075118720833937628416751

Graph of the $Z$-function along the critical line